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Mirrors > Home > MPE Home > Th. List > tpostpos2 | Structured version Visualization version GIF version |
Description: Value of the double transposition for a relation on triples. (Contributed by Mario Carneiro, 16-Sep-2015.) |
Ref | Expression |
---|---|
tpostpos2 | ⊢ ((Rel 𝐹 ∧ Rel dom 𝐹) → tpos tpos 𝐹 = 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tpostpos 7915 | . 2 ⊢ tpos tpos 𝐹 = (𝐹 ∩ (((V × V) ∪ {∅}) × V)) | |
2 | relrelss 6127 | . . . 4 ⊢ ((Rel 𝐹 ∧ Rel dom 𝐹) ↔ 𝐹 ⊆ ((V × V) × V)) | |
3 | ssun1 4151 | . . . . . 6 ⊢ (V × V) ⊆ ((V × V) ∪ {∅}) | |
4 | xpss1 5577 | . . . . . 6 ⊢ ((V × V) ⊆ ((V × V) ∪ {∅}) → ((V × V) × V) ⊆ (((V × V) ∪ {∅}) × V)) | |
5 | 3, 4 | ax-mp 5 | . . . . 5 ⊢ ((V × V) × V) ⊆ (((V × V) ∪ {∅}) × V) |
6 | sstr 3978 | . . . . 5 ⊢ ((𝐹 ⊆ ((V × V) × V) ∧ ((V × V) × V) ⊆ (((V × V) ∪ {∅}) × V)) → 𝐹 ⊆ (((V × V) ∪ {∅}) × V)) | |
7 | 5, 6 | mpan2 689 | . . . 4 ⊢ (𝐹 ⊆ ((V × V) × V) → 𝐹 ⊆ (((V × V) ∪ {∅}) × V)) |
8 | 2, 7 | sylbi 219 | . . 3 ⊢ ((Rel 𝐹 ∧ Rel dom 𝐹) → 𝐹 ⊆ (((V × V) ∪ {∅}) × V)) |
9 | df-ss 3955 | . . 3 ⊢ (𝐹 ⊆ (((V × V) ∪ {∅}) × V) ↔ (𝐹 ∩ (((V × V) ∪ {∅}) × V)) = 𝐹) | |
10 | 8, 9 | sylib 220 | . 2 ⊢ ((Rel 𝐹 ∧ Rel dom 𝐹) → (𝐹 ∩ (((V × V) ∪ {∅}) × V)) = 𝐹) |
11 | 1, 10 | syl5eq 2871 | 1 ⊢ ((Rel 𝐹 ∧ Rel dom 𝐹) → tpos tpos 𝐹 = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 Vcvv 3497 ∪ cun 3937 ∩ cin 3938 ⊆ wss 3939 ∅c0 4294 {csn 4570 × cxp 5556 dom cdm 5558 Rel wrel 5563 tpos ctpos 7894 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-fv 6366 df-tpos 7895 |
This theorem is referenced by: 2oppchomf 16997 mattpostpos 21066 |
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